ĐKXĐ: \(x\ge0,x\ne9\)

a) \(P=\frac{3\sqrt{x}+2}{\sqrt{x}+1}+\frac{2\sqrt{x}+3}{\sqrt{x}-3}-\frac{3\left(3\sqrt{x}-5\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x-3}\right)}\)

\(=\frac{\left(3\sqrt{x}+2\right)\left(\sqrt{x}-3\right)+\left(2\sqrt{x}-3\right)\left(\sqrt{x}+1\right)+3\left(3\sqrt{x}-5\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\)

\(=\frac{3x-9\sqrt{x}+2\sqrt{x}-6+2x+2\sqrt{x}-3\sqrt{x}-3-9\sqrt{x}+15}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\)

\(=\frac{5x-17\sqrt{x}+6}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\)

\(=\frac{5x-15\sqrt{x}-2\sqrt{x}+6}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\)

\(=\frac{\left(5\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}=\frac{5\sqrt{x}-2}{\sqrt{x}+1}\)

b) Ta có: \(x=4+2\sqrt{3}=\left(\sqrt{3}+1\right)^2\Rightarrow\sqrt{x}=\sqrt{3}+1\)

Do đó: \(P=\frac{5\left(\sqrt{3}+1\right)-2}{\left(\sqrt{3}+1\right)+1}=\frac{5\sqrt{3}+3}{\sqrt{3}+2}=\frac{\left(5\sqrt{3}+3\right)\left(2-\sqrt{3}\right)}{\left(\sqrt{3}+2\right)\left(2-\sqrt{3}\right)}=7\sqrt{3}-9\)

c) Ta có \(P=\frac{5\sqrt{x}-2}{\sqrt{x}+1}=\frac{5\sqrt{x}+5-7}{\sqrt{x}+1}\)

\(P=5-\frac{7}{\sqrt{x}+1}\)

Vì \(\frac{7}{\sqrt{x}+1}>0\)nên \(P\)có giá trị nhỏ nhất khi và chỉ khi \(\frac{7}{\sqrt{x}+1}\)lớn nhất

\(\Leftrightarrow\sqrt{x}+1\)nhỏ nhất \(\Leftrightarrow x=0\)

Khi đó min_P=5-7=-2

Đặt \(\sqrt{x+3}=a\Rightarrow x+3=a^2\)

\(\Rightarrow3=a^2-x\)

Ta có phương trình mới là :

 \(\left(2x-1\right)a=x^2+a^2-x \)

\(\Leftrightarrow x^2+a^2-x-2ax+a=0\)

\(\Leftrightarrow\left(a^2-2ax+x^2\right)+\left(a-x\right)=0\)

\(\Leftrightarrow\left(a-x\right)^2-\left(a-x\right)=0\)

\(\Leftrightarrow\left(a-x\right)\left(a-x-1\right)=0\)

Với \(a-x=0\Leftrightarrow a=x\)

\(\Leftrightarrow\sqrt{x+3}=x\)  \(\left(x\ge-3\right)\)

\(\Leftrightarrow x^2-x-3=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=\frac{1+\sqrt{13}}{2}\\x=\frac{1-\sqrt{13}}{2}\end{cases}}\)t/m

Với  \(a-x-1=0\)\(\Leftrightarrow\sqrt{x+3}=x+1\)

\(\Leftrightarrow x^2+x-2=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=1\\x=-2\end{cases}}\)

Vậy...

\(đk:-5\le x\le3\)

\(\sqrt{x+5}+\sqrt{3-x}=2\left(\sqrt{15-2x-x^2}+1\right)\)                     (1)

\(\Leftrightarrow\sqrt{x+5}+\sqrt{3-x}=2\left(\sqrt{\left(x+5\right)\left(3-x\right)}+1\right)\)

\(\Leftrightarrow\sqrt{x+5}=\sqrt{3-x}=2\sqrt{\left(x+5\right)\left(3-x\right)}+2\)

đặt \(\sqrt{x+5}+\sqrt{3-x}=t\)   (đk t > 0)

\(\Leftrightarrow t^2=x+5+2\sqrt{\left(x+5\right)\left(3-x\right)}+3-x\)

\(\Leftrightarrow t^2=8+2\sqrt{\left(x+5\right)\left(3-x\right)}\)

\(\Leftrightarrow t^2=6+\left(2+2\sqrt{\left(x+5\right)\left(3-x\right)}\right)\)    và (1)

\(\Rightarrow t=t^2-6\)

\(\Leftrightarrow t^2-t-6=0\)

\(\Leftrightarrow\orbr{\begin{cases}t=-2\left(loai\right)\\t=3\end{cases}}\)    

ta có : \(8+2\sqrt{\left(x+5\right)\left(3-x\right)}=3^2=9\)

\(\Leftrightarrow2\sqrt{\left(x+5\right)\left(3-x\right)}=1\)

\(\Leftrightarrow\sqrt{\left(x+5\right)\left(3-x\right)}=\frac{1}{2}\)

\(\Leftrightarrow\left(x+5\right)\left(3-x\right)=\frac{1}{4}\)

\(\Leftrightarrow59-8x-4x^2=0\)

\(\Leftrightarrow4x^2+8x+4-63=0\)

\(\Leftrightarrow4\left(x+1\right)^2=63\) \(\Leftrightarrow\left(x+1\right)^2=\frac{63}{4}\Leftrightarrow x+1=\pm\sqrt{\frac{63}{4}}\)

\(\Leftrightarrow x=\pm\sqrt{\frac{63}{4}}-1\left(tm\right)\)

ĐKXĐ : \(x\ne-\frac{1}{3}\)

Ta có : \(\sqrt{x^2+x+2}=\frac{3x^2+3x+2}{3x+1}\)

\(\Leftrightarrow\sqrt{x^2+x+2}-2=\frac{3x^2+3x+2}{3x+1}-2\)

\(\Leftrightarrow\frac{x^2+x+2-4}{\sqrt{x^2+x+2}+2}=\frac{3x^2+3x+2-6x-2}{3x+1}\)

\(\Leftrightarrow\frac{x^2+x-2}{\sqrt{x^2+x+2}+2}=\frac{3x^2-3x}{3x+1}\)

\(\Leftrightarrow\frac{\left(x-1\right)\left(x+2\right)}{\sqrt{x^2+x+2}+2}-\frac{3x\left(x-1\right)}{3x+1}=0\)

\(\Leftrightarrow\left(x-1\right)\left[\frac{x+2}{\sqrt{x^2+x+2}+2}-\frac{3x}{3x+1}\right]=0\)

\(\Leftrightarrow x=1\)( Thỏa mãn )

\(\left(a^2+b^2\right)\left(c^2+d^2\right)=5.5\)

\(\Leftrightarrow a^2c^2+a^2d^2+b^2c^2+b^2d^2=25\)

\(\Leftrightarrow\left(a^2c^2+2abcd+b^2d^2\right)+\left(a^2d^2-2abcd+b^2c^2\right)=25\)

\(\Leftrightarrow\left(ac+bd\right)^2+\left(ad-bc\right)^2=25\)

mà ac + bd = 3

\(\Leftrightarrow\left(ad-bc\right)^2=25-3^2=16\)

\(\Leftrightarrow\orbr{\begin{cases}ad-bc=4\\ad-bc=-4\end{cases}}\)

a)\(\frac{1}{\left(n+1\right).\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)^2.n-n^2\left(n+1\right)}\)

\(=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)

b)\(S=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{399}}-\frac{1}{\sqrt{400}}\)

\( S=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{400}}=1-\frac{1}{20}=\frac{19}{20}\)

\(a,\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n}\cdot\sqrt{n+1}\left(\sqrt{n}+\sqrt{n+1}\right)}\)

\(=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}\cdot\sqrt{n+1}\left(\sqrt{n}+\sqrt{n+1}\right)\left(\sqrt{n+1}-\sqrt{n}\right)}\)

\(=\frac{\sqrt{n-1}-\sqrt{n}}{\sqrt{n}\cdot\sqrt{n+1}\left(n+1-n\right)}\)

\(=\frac{\sqrt{n-1}-\sqrt{n}}{\sqrt{n}\cdot\sqrt{n+1}}=\frac{\sqrt{n-1}}{\sqrt{n}\cdot\sqrt{n+1}}-\frac{\sqrt{n}}{\sqrt{n}\cdot\sqrt{n+1}}\)

\(=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)

b, \(S=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{3\sqrt{4}+4\sqrt{3}}+...+\frac{1}{400\sqrt{399}+399\sqrt{400}}\)

\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{399}}-\frac{1}{\sqrt{400}}\)

\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{400}}=1-\frac{1}{20}=\frac{19}{20}\)

\(\left(x+y\right)^2+3x+y+1=z^2\)với x,y,z nguyên dương \(\Rightarrow z^2>\left(x+y\right)^2\)

\(\left(x+y\right)^2+3x+y+1=\left(x+y+2\right)^2-x-3y-3=z^2\)với x,y,z nguyên dương \(\Rightarrow z^2< \left(x+y+2\right)^2\)

Vậy \(z^2\)là số chính phương ở giữa 2 số chính phương khác là \(\left(x+y\right)^2\)và \(\left(x+y+2\right)^2\)

\(\Rightarrow z^2=\left(x+y+1\right)^2\Leftrightarrow\orbr{\begin{cases}x+y=1-z\left(1\right)\\x+y=z-1\left(2\right)\end{cases}}\)

Xét (1): \(x+y=1-z>0\Rightarrow z< 1\Leftrightarrow z=0\)Vì 0 không là số nguyên dương nên (1) vô nghiệm.

Xét (2): \(x+y=z-1\)lúc này pt có vô số nghiệm nguyên dương (x;y;z), x>0, y>0, z>1

Bạn hỏi tự vẽ hình nhá

a) Kẻ \(ME\perp AD,MF\perp BC,MG\perp AB,MH\perp CD\)

\(MA^2+MC^2=MB^2+MD^2\)( cùng bằng \(ME^2+MG^2+MF^2+MH^2\))

b) Chứng mih tương tự=>kết quả không đổi. 

Ta có: \(MA^2+MC^2=MB^2+MD^2\)(cùng bằng \(ME^2=AE^2+MF^2+CF^2\))

Vậy khi điểm M nằm ngoài hình chữ nhật ABCD thì đẳng thức ở câu a) vẫn đúng.

Ta có \(B=\frac{x^2-4x+5}{2}=\frac{x^2-4x+4}{2}+\frac{1}{2}=\left(x-2\right)^2.\frac{1}{2}+\frac{1}{2}\ge\frac{1}{2}\)

Dấu "=" xảy ra <=> x - 2 = 0

=> x = 2

Vậy Min B = 1/2 <=> x = 2

\(B=\frac{x^2-4x+5}{2}=\frac{x^2-4x+4+1}{2}=\frac{\left(x-2\right)^2}{2}+\frac{1}{2}\)

Vì \(\left(x-2\right)^2\ge0\forall x\)\(\Rightarrow\frac{\left(x-2\right)^2}{2}\ge0\forall x\)

\(\Rightarrow\frac{\left(x-2\right)^2}{2}+\frac{1}{2}\ge\frac{1}{2}\forall x\)

Dấu " = " xảy ra \(\Leftrightarrow x-2=0\)\(\Leftrightarrow x=2\)

Vậy \(minB=\frac{1}{2}\)\(\Leftrightarrow x=2\)